Main content

### Course: Integrated math 3 > Unit 6

Lesson 8: Graphs of logarithmic functions# Graphical relationship between 2ˣ and log₂(x)

Sal graphs y=2ˣ and y=log₂(x) on the same coordinate plane, showing how they relate as graphs of inverse functions. Created by Sal Khan.

## Want to join the conversation?

- How were the exponential and logarithm functions invented?(16 votes)
- I've read that Leonhard Euler "discovered" log functions. I'm not sure if that means he recognized the pattern, or he just named them "logarithmic functions"(16 votes)

- How did you get 1(7 votes)
- Any number raised to power 0 = 1.

For example, 1^0 = 1

5^0 = 1

1000^0 =1.

This is one property of exponents. With y=2^x, if you check what is the value of y if x=0, then y=2^0=1. From this, you can also conclude that for any basis c (y=c^x), when x=0, => y=1.(22 votes)

- At4:40, how did someone found out all this?(7 votes)
- As Sal says, exponential functions and logarithmic functions are inverses so they appear as reflections on the graph. Basically, the x values and y values are swapped.(8 votes)

- How do I find the Asymptote of a logarithmic function? Is there any formula for this?(6 votes)
- Logarithmic functions have vertical asymptotes, so follow the graph and see where it won't go past a line (ex. x equals 0 or the y-axis for the log in this video. Also, since log graphs are inverses of the exp graphs, an exp graph with a horizontal aymptote of the x axis would have a log asymptote of the y axis (I believe).(9 votes)

- How do you simplify log functions. or express them when (log...) is the exponent?(4 votes)
- You can think of log functions in the form of 'y=logbx', with b as the base, you switch the positions of the letters, making it 'x=b^y'. When switching from log to exponential forms, make it 'x=', make 'b' the base of the other side, and make 'y' the exponent.

I know it's been 2 years, but I hope this helps.(5 votes)

- Are logarithmic functions exponential?(3 votes)
- No. The logarithmic function is the inverse function of the exponential function. This is means that if a^x = b (exponential), then log base a (b) = x. (logarithmic). Therefore, exponential and logarithmic functions are not the same.(6 votes)

- How do you find the Asymptote of a logarithmic function?

I'll greatly appreciate any help!(4 votes)- You should know the asymptote of the parent function y=log(x) is the y axis (x=0). Just as with other functions, to move the asymptote, you have to add or subtract something to the x. If you do y=log(x-5) that would move the asymptote 5 units to the right, so at x=5. y=log(x+3) moves it to left 3 units at x=-3. Note that the formula for moving is y=a log(x-h) + k, so h moves opposite its sign.(3 votes)

- How would you graph f=-log_2_(x)-4? I think it would be different from how exponential functions work.(3 votes)
- If it helps, you could rearrange the equation to be y+4=log_2(x), and then x=2^(y+4), then graph it as a function in terms of y.

If you don’t want to do the rearranging, plug in a few points (at x=1 and x=2 would be easy to calculate in your head) and then draw a smooth curve. The method is similar to an exponential function, but it does look different.

Let me know if you have further questions.(5 votes)

- Can logarithmic functions we written in the form y = a * (log_b(x-h)) + k just like parabolic functions can be written as y = a(x-h)^2 + k?(4 votes)
- isnt that logarithm picked arbitrarily?(3 votes)
- Not sure what you mean, there is a definite pattern to go between log notation and exponential notation, they are inverse functions of each other, so this pattern in not arbitrary.(2 votes)

## Video transcript

Voiceover:What I want to do in this video is graph up a classic exponential function and then graph a related
logarithmic function and see how the two are related visually. The two things I'm going to graph are y is equal to two to the x power and y is equal to the log base two of x. I encourage you to pause the video, make a table for each of them and try to graph them
on the same graph paper. See how they are related and if you see how they're related, think about why they are related that way. Let's first start with y
equals two to the x power. I'm going to make a little table here, different x values and the
corresponding y values. x and y, we can start with negative two, negative one, zero, one, two, three. In each case y is going to
be two raised to these power. Two to the negative two power is going to be 1/4. Two to the negative one power is 1/2. Two to the zero power is one. Two to the first power is two. Two to the second power is four. Two to the third power is eight. Let's graph that. Two to the third power is eight. Two to the second power is four. Two to the first power is two. Two to the zeroth power is one. Two to the negative one power is 1/2. Two to the negative two power is 1/4. Even the two to the negative third power is going to be 1/8, so it's going to look something like this. The graph is going to
look something like this right over here. It's kind of your classic, sometimes this will be called
your exponential hockey stick because it kind of looks
like a hockey stick where it just kind of starts kind of slow and just oohh bam, shoots straight up. Notice as we go to the left as x becomes more and
more and more negative our value approaches zero
but never quite gets there. If we have two to the
negative one millionth power it's going to be a
very, very small number, very, very close to zero but it's not going to be quite zero. We're going to have a
horizontal asymptote at y is equal to zero or the x-axis is a horizontal asymptote. Fair enough. Now let's graph y is equal
to log base two of x. Before I graph that, let's
just think about another way of representing it. This literally says, for any x, what power, what exponent
y if I raise two to that would give me x. This is an equivalent statement as saying two to the
y power is equal to x. If you notice, what we've done here between these two things you're essentially just
switching the x's and the y's. Here's two to the x power is equal to y. Here's two to the y power is equal to x. Really this and this you've
swapped the x's and the y's. What we will see is
that we can essentially swap these two columns. x and y, so let me just do 1/4, 1/2, one, two, four, and eight. Here now we're saying if x is 1/4, what power do we have to raise two to, to get to 1/4. We have to raise it to
the negative two power. Two to the negative one
power is equal to 1/2. Two to the zero power is equal to one. Two to the first power is equal to two. Two to the second power is equal to four. Two to the third power is equal to eight. Notice all we did, as we essentially swapped
these two columns, so let's graph this. When x is equal to 1/4, y
is equal to negative two. When x is 1/2, y is equal to negative one. When x is one, y is zero. When x is two, y is one. When x is four, y is two. When x is eight, y is three. It's going to look like this. Notice, I think you
might already be seeing a pattern right over here. These two graphs are essentially the reflections of each other. What would you have to reflect about to get these two? Well you'd have to reflect
about y is equal to x. If you swap the x's and the y's, another way to think about, if you swap the axis you
would get the other graph. It's essentially what we're doing. Notice it's symmetric about that line and that's because these are essentially the inverse functions of each other. One way to think about it is we swapped the x's and y's. Just as this, as x becomes more and more and more and more negative you see y approaching zero. Here you see is y is becoming
more and more negative as x is approaching zero, or you could say as x approaches zero y becomes more and more and more negative. The whole point of this is just to give you an appreciation for the relationship between
an exponential function and a logarithmic function. They're essentially
inverses of each other. You see that in the graphs, they're reflections of each other about the line y is equal to x.